void	nag_wilcoxon_test (Integer n, const double x[], double median, Nag_TailProbability tail, Nag_IncSignZeros zeros, double w, double z, double p, Integer non_zero, NagError *fail)

3 Description

The Wilcoxon one sample signed rank test may be used to test whether a particular sample came from a population with a specified median. It is assumed that the population distribution is symmetric. The data consist of a single sample of

n

observations denoted by

x_{1}, x_{2}, \dots, x_{n}

. This sample may arise from the difference between pairs of observations from two matched samples of equal size taken from two populations, in which case the test may be used to test whether the median of the first population is the same as that of the second population.

The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the median is equal to some given value

(X_{med})

, and this is to be tested against an alternative hypothesis

H_{1}

which is

$H_{1}$ : population median $\neq X_{med}$ ; or
$H_{1}$ : population median $> X_{med}$ ; or
$H_{1}$ : population median $< X_{med}$ ,

using a two tailed, upper tailed or lower tailed probability respectively. You select the alternative hypothesis by choosing the appropriate tail probability to be computed (see the description of argument tail in Section 5).

The Wilcoxon test differs from the Sign test (see nag_sign_test (g08aac)) in that the magnitude of the scores is taken into account, rather than simply the direction of such scores.

The test procedure is as follows:

(a)	For each $x_{i}$ , for $i = 1, 2, \dots, n$ , the signed difference $d_{i} = x_{i} - X_{med}$ is found, where $X_{med}$ is a given test value for the median of the sample.
(b)	The absolute differences $\|d_{i}\|$ are ranked with rank $r_{i}$ and any tied values of $\|d_{i}\|$ are assigned the average of the tied ranks. You may choose whether or not to ignore any cases where $d_{i} = 0$ by removing them before or after ranking (see the description of the argument zeros in Section 5).
(c)	The number of nonzero $d_{i}$ 's is found.
(d)	To each rank is affixed the sign of the $d_{i}$ to which it corresponds. Let $s_{i} = sign (d_{i}) r_{i}$ .
(e)	The sum of the positive-signed ranks, $W = \sum_{s_{i} > 0} s_{i} = \sum_{i = 1}^{n} \max (s_{i}, 0.0)$ , is calculated.

nag_wilcoxon_test (g08agc) returns:

(a)

The test statistic

W

;

(b)

The number

n_{1}

of nonzero

d_{i}

's;

(c)

The approximate Normal test statistic

z

, where

z = \frac{(W - \frac{n_{1} (n_{1} + 1)}{4}) - sign (W - \frac{n_{1} (n_{1} + 1)}{4}) \times \frac{1}{2}}{\sqrt{\frac{1}{4} \sum_{i = 1}^{n} s_{i}^{2}}}

(d)

The tail probability,

p

, corresponding to

W

, depending on the choice of the alternative hypothesis,

H_{1}

n_{1} \leq 80

p

is computed exactly; otherwise, an approximation to

p

is returned based on an approximate Normal statistic corrected for continuity according to the tail specified.

The value of

p

can be used to perform a significance test on the median against the alternative hypothesis. Let

α

be the size of the significance test (that is,

α

is the probability of rejecting

H_{0}

when

H_{0}

is true). If

p < α

then the null hypothesis is rejected. Typically

α

might be 0.05 or 0.01.

4 References

Conover W J (1980) Practical Nonparametric Statistics Wiley

Neumann N (1988) Some procedures for calculating the distributions of elementary nonparametric teststatistics Statistical Software Newsletter 14(3) 120–126

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

5 Arguments

1: $n$ – IntegerInput

On entry: the size of the sample,

n

Constraint:

n \geq 1

2: $x [n]$ – const doubleInput

On entry: the sample observations,

x_{1}, x_{2}, \dots, x_{n}

3: $median$ – doubleInput

On entry: the median test value,

X_{med}

4: $tail$ – Nag_TailProbabilityInput

On entry: indicates the choice of tail probability, and hence the alternative hypothesis.

$tail = Nag_TwoTail$: A two tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $\neq X_{med}$ .
$tail = Nag_UpperTail$: An upper tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $> X_{med}$ .
$tail = Nag_LowerTail$: A lower tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $< X_{med}$ .

Constraint:

tail = Nag_TwoTail

Nag_UpperTail

Nag_LowerTail

5: $zeros$ – Nag_IncSignZerosInput

On entry: indicates whether or not to include the cases where

d_{i} = 0.0

in the ranking of the

d_{i}

's.

$zeros = Nag_IncSignZerosY$: All $d_{i} = 0.0$ are included when ranking.
$zeros = Nag_IncSignZerosN$: All $d_{i} = 0.0$ , are ignored, that is all cases where $d_{i} = 0.0$ are removed before ranking.

Constraint:

zeros = Nag_IncSignZerosY

Nag_IncSignZerosN

6: $w$ – double *Output

On exit: the Wilcoxon rank sum statistic,

W

, being the sum of the positive ranks.

7: $z$ – double *Output

On exit: the approximate Normal test statistic,

z

, as described in Section 3.

8: $p$ – double *Output

On exit: the tail probability,

p

, as specified by the argument tail.

9: $non_zero$ – Integer *Output

On exit: the number of nonzero

d_{i}

's,

n_{1}

10: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument tail had an illegal value.

On entry, argument zeros had an illegal value.
NE_G08AG_SAMP_IDEN: The whole sample is identical to the given median test value.
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

The approximation used to calculate

p

when

n_{1} > 80

will return a value with a relative error of less than 10 percent for most cases. The error may increase for cases where there are a large number of ties in the sample.

8 Parallelism and Performance

nag_wilcoxon_test (g08agc) is not threaded in any implementation.

9 Further Comments

The time taken by nag_wilcoxon_test (g08agc) increases with

n_{1}

, until

n_{1} > 80

, from which point on the approximation is used. The time decreases significantly at this point and increases again modestly with

n_{1}

for

n_{1} > 80

10 Example

The example program performs the Wilcoxon signed rank test on two matched samples of size 8, taken from two populations. The distribution of the differences between pairs of observations from the two populations is assumed to be symmetric. The test is used to test whether the medians of the two distributions of the populations are equal or not. The test statistic, the approximate Normal statistic and the two tailed probability are computed and printed.

NAG Library Function Documentnag_wilcoxon_test (g08agc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_wilcoxon_test (g08agc)